This paper considers the minimization of a general objective function f(X) over the set of rectangular n $\times$ m matrices that have rank at most r. To reduce the computational burden, we factorize the variable X into a product of two smaller matrices and optimize over these two matrices instead of X. Despite the resulting nonconvexity, recent studies in matrix completion and sensing have shown that the factored problem has no spurious local minima and obeys the so-called strict saddle property (the function has a directional negative curvature at all critical points but local minima). We analyze the global geometry for a general and yet well-conditioned objective function f(X) whose restricted strong convexity and restricted strong smoothness constants are comparable. In particular, we show that the reformulated objective function has no spurious local minima and obeys the strict saddle property. These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) can provably solve the factored problem with global convergence.

Due to its wide applications, remote sensing (RS) image scene classification has attracted increasing research interest. When each category has a sufficient number of labeled samples, RS image scene classification can be well addressed by deep learning. However, in the RS big data era, it is extremely difficult or even impossible to annotate RS scene samples for all the categories in one time as the RS scene classification often needs to be extended along with the emergence of new applications that inevitably involve a new class of RS images. Hence, the RS big data era fairly requires a zero-shot RS scene classification (ZSRSSC) paradigm in which the classification model learned from training RS scene categories obeys the inference ability to recognize the RS image scenes from unseen categories, in common with the humans’ evolutionary perception ability. Unfortunately, zero-shot classification is largely unexploited in the RS field. This article proposes a novel ZSRSSC method based on locality-preservation deep cross-modal embedding networks (LPDCMENs). The proposed LPDCMENs, which can fully assimilate the pairwise intramodal and intermodal supervision in an end-to-end manner, aim to alleviate the problem of class structure inconsistency between two hybrid spaces (i.e., the visual image space and the semantic space). To pursue a stable and generalization ability, which is highly desired for ZSRSSC, a set of explainable constraints is specially designed to optimize LPDCMENs. To fully verify the effectiveness of the proposed LPDCMENs, we collect a new large-scale RS scene data set, including the instance-level visual images and class-level semantic representations (RSSDIVCS), where the general and domain knowledge is exploited to construct the class-level semantic representations. Extensive experiments show that the proposed ZSRSSC method based on LPDCMENs can obviously outperform the state-of-the-art methods, and the domain knowledge further improves the performance of ZSRSSC compared with the general knowledge. The collected RSSDIVCS will be made publicly available along with this article.

The low-rankness property of the Hankel matrix formulated from the clean seismic data corresponding to a few number of linear events has been successively leveraged in many low-rank (LR) approximation methods for seismic data denoising. The common scheme in these rank-reduction methods is to compute the best LR approximation of the formulated Hankel matrix and then obtain the denoised data from the LR matrix. However, without utilizing the Hankel structure when computing the LR approximation, if we rearrange the denoised data into a Hankel matrix, it is in general not exactly LR as expected. In this paper, we propose a Hankel LR (HLR) approximation method to simultaneously exploit both the Hankel structure and the LR property underlying the clean seismic data. The formulated HLR approximation problem is solved by an alternating-minimization-based algorithm. We provide rigorously convergence analysis of the proposed algorithm. The superior performance of the proposed HLR approximation method is demonstrated on both synthetic and field seismic data.

This paper considers general rank-constrained optimization problems that minimize a general objective function ${f}( {X})$ over the set of rectangular ${n}\times {m}$ matrices that have rank at most r. To tackle the rank constraint and also to reduce the computational burden, we factorize $ {X}$ into $ {U} {V} ^{\mathrm {T}}$ where $ {U}$ and $ {V}$ are ${n}\times {r}$ and ${m}\times {r}$ matrices, respectively, and then optimize over the small matrices $ {U}$ and $ {V}$ . We characterize the global optimization geometry of the nonconvex factored problem and show that the corresponding objective function satisfies the robust strict saddle property as long as the original objective function f satisfies restricted strong convexity and smoothness properties, ensuring global convergence of many local search algorithms (such as noisy gradient descent) in polynomial time for solving the factored problem. We also provide a comprehensive analysis for the optimization geometry of a matrix factorization problem where we aim to find ${n}\times {r}$ and ${m}\times {r}$ matrices $ {U}$ and $ {V}$ such that $ {U} {V} ^{\mathrm {T}}$ approximates a given matrix $ {X}^\star $ . Aside from the robust strict saddle property, we show that the objective function of the matrix factorization problem has no spurious local minima and obeys the strict saddle property not only for the exact-parameterization case where $\mathrm {rank}( {X}^\star) = {r}$ , but also for the over-parameterization case where $\mathrm {rank}( {X}^\star) < {r}$ and the under-parameterization case where $\mathrm {rank}( {X}^\star) > {r}$ . These geometric properties imply that a number of iterative optimization algorithms (such as gradient descent) converge to a global solution with random initialization.

Coherent imaging systems are greatly affected by speckle noise, which makes visual analysis and features extraction a difficult task. In this paper, we propose a speckle suppression algorithm based on weighted nuclear norm minimization (WNNM) and Grey theory. First, we use logarithmic transformation to the noisy images such that the speckle noise is transformed into additive noise. Second, by matching the local blocks based on Grey theory, we will get approximate low-rank matrices grouped by the similar blocks of the reference patches. We then estimate the noise variance of the noisy images with the wavelet transform. Finally, we use WNNM method to denoise the image. The results show that our algorithm not only effectively improves the visual effect of the denoised image and preserves the local structure of the image better but also improves the objective indexes values of the denoised image.